3.24 \(\int \sqrt {\cos (c+d x)} (3-5 \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=21 \[ -\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d} \]

[Out]

-2*cos(d*x+c)^(3/2)*sin(d*x+c)/d

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {3011} \[ -\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Cos[c + d*x]]*(3 - 5*Cos[c + d*x]^2),x]

[Out]

(-2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/d

Rule 3011

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*Cos[e
+ f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[A*(m + 2) + C*(m +
1), 0]

Rubi steps

\begin {align*} \int \sqrt {\cos (c+d x)} \left (3-5 \cos ^2(c+d x)\right ) \, dx &=-\frac {2 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 23, normalized size = 1.10 \[ -\frac {\sin (2 (c+d x)) \sqrt {\cos (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Cos[c + d*x]]*(3 - 5*Cos[c + d*x]^2),x]

[Out]

-((Sqrt[Cos[c + d*x]]*Sin[2*(c + d*x)])/d)

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fricas [A]  time = 0.46, size = 19, normalized size = 0.90 \[ -\frac {2 \, \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-5*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

-2*cos(d*x + c)^(3/2)*sin(d*x + c)/d

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} \sqrt {\cos \left (d x + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-5*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate(-(5*cos(d*x + c)^2 - 3)*sqrt(cos(d*x + c)), x)

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maple [B]  time = 0.62, size = 99, normalized size = 4.71 \[ -\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3-5*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x)

[Out]

-4*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2
*d*x+1/2*c)^2)^(1/2)*(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/sin(1/2*d*x+1/2*c)/d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int {\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} \sqrt {\cos \left (d x + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-5*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

-integrate((5*cos(d*x + c)^2 - 3)*sqrt(cos(d*x + c)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ -\int \sqrt {\cos \left (c+d\,x\right )}\,\left (5\,{\cos \left (c+d\,x\right )}^2-3\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-cos(c + d*x)^(1/2)*(5*cos(c + d*x)^2 - 3),x)

[Out]

-int(cos(c + d*x)^(1/2)*(5*cos(c + d*x)^2 - 3), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-5*cos(d*x+c)**2)*cos(d*x+c)**(1/2),x)

[Out]

Timed out

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